Melancholiastatesintheclimatesystem:exploringglobalinstabili5esandcri5cal

transi5onsValerioLucarini

UniversityofReading/UniversityofHamburg

Thanks:T.Bodai,R.Boschi,H.Dijkstra,B.Eckhardt,C.Grebogi,A.Gritsun,F.Lunkeit,E.OF,F.Ragone,A.Tantet,J.Yorke

LesHouchesAugust3rd2017

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Whatcomesnext•  Reading,Aug-Sept2017

–  CliMathNetConference(withAshwin,Bodai,Broecker,Fowler,Freitag,Kuna,Neves,ScoF,Shepherd,Williams)

•  ICTP,May2018–  AdvancedWorkshoponNonequilibriumSystemsinPhysics,Geosciences,andLifeSciences(withBouchet,Ruffo,Gallavo^,Gambassi)

•  LesHouches,Feb-March2019–  PhysicsandMathema`csofTurbulentFlowsatDifferentScales(withDubrulle,Faranda,Wouters,GoFwald)

•  Inst.Poincare,Autumn2019–  Themathema`csofclimateandtheenvironment(withGhil,Chekroun,Klein,LeTreut,Speich)

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Melancholia(2011,VonTrier)

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•  CloseencounterbtwEarthandbrowndwarf–Tensiongrows

•  Veryclosetothe`ppingpoint-Tensionpeaks

•  UnavoidableImpact–Tensionisreleased

Mo`va`ons:Basics•  Understandinghowasystemrespondstoperturba`onsisacentralareaofresearchinnaturalandmathema`calsciences–  Robustnessofthesystem?–  Smoothresponse?–  Cri`calTransi`ons?

•  GroundbreakingworkbyKubo(1957)–  Responsetheoryforsta`s`calphysicalsystems–  Onlyfornear-equilibrium(canonicalensemble)– Mathema`callyandphysicallynon-rigorous,manycri`cisms–  Acous`cs,Op`cs,etc.basedonKubo’sresults–  Fluctua`on-Dissipa`onTheorem:dic`onarybetweenforcedandfreefluctua`ons

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FromSmoothResponse…•  Ruelle(‘90s):rigorousresponsetheoryforsmoothobservablesofAxiomAsystems(eq.&noneq.!)– UsualFDTdoesnotapplyfornonequilibriumsystems:unstablevsstabledirec`onsintangentspace

–  Cri`calTransi`onsaslossofsmoothnessinresponse–  Theoryusefultoperformpredic`onsbuthardtoconstructresponseoperator!

•  Liverani,Baladi,Dolgopyat,..(‘00s)–  Responsetheoryderivedusingtransferoperatorapproach–  Changeintheinvariantmeasurevschangeinobservables–  BeyondAxiomAsystems

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…toCri`calTransi`ons•  Catastrophetheory(’60s,Thom,Arnold):comprehensiveviewofbifurca`onsin“rela`velysimplesystems’

•  Mul`stabilityincomplexsystems&hysteresis•  Definingtheboundariesbetweenthebasins•  Ashwinetal.2012:Parameter-,rate-,andnoise-induced`pping

•  Freidlin-Wentzelltheory(’70s)basedonLargeDevia`onsTheory(‘60s):generallawsfornoise-inducedescapefrombasinsofaFrac`on

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WhyClimateisrelevant•  Theclimateisanonequilibriumsystemwhoseevolu`onisdrivenbyinhomogeneousabsorp`onofsolarradia`on

•  Theclimatefeaturesvariabilityofavastrangeofspa`alandtemporalscales

•  Understandingtheclimateresponsetoperturba`onisgreatscien`ficchallenge– Anthropogenicclimatechange–  Paleoclimate→Life–  PlanetaryScience→Habitability

•  Smoothvs.nonsmoothresponse–  Climatechange,climatesurprises,climate`ppingpoints

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•  Ver`cal

•  Horizontal

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Nonequilibrium

Anextremelynon-idealengine

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T1 T2

Qin Qout

dissipation

Q1 Q2

W

Irreversible heat transport

•  Efficiency•  Energytransforma`on•  EntropyProduc`on

Scales of Motions Smagorinsky/Stommel

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ClimateResponse

A.  Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?

B.  Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters

C.  Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor

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14

A

J.Stat.Phys.166,1036-1064(2017)

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•  Perturba`ontoAxiomA:•  Changeinexpecta`onvalueofasmoothΦ:

•  Linearterm:

•  LinearGreen:•  Linearsuscept:obeysKKrela`ons

Construc5ngthe5me-dependentmeasureviaRuelleResponseTheory

Φ1( ) t( ) = dσ∫ GΦ

(1) σ( )e t −σ( )GΦ(1) t( ) = ρ0 dx( )∫ Θ t( )X ⋅∇StΦ

χΦ(1) ω( ) = dt∫ exp iωt[ ]GΦ

(1) t( )

Φ1( ) ω( ) = χΦ

(1) ω( )e ω( )

!x = !F x, t( )= F x( )+εe t( )X x( )

Φ t( ) = Φ0+ ε k Φ

(k ) t( )k=1

∞

∑ PullbackaFractor

ModelStarterand

GraphicUserInterface

SpectralAtmospheremoistprimi`veequa`ons

onσlevels

Sea-Icethermodynamic

TerrestrialSurface:fivelayersoilplussnow

Vegeta`ons(Simba,V-code,

Koeppen)

Oceans:LSG,mixedlayer,orclimatol.SST

PLASIM: An efficient Climate Model

Keyfeatures• portable• fast• opensource• parallel• modular• easytouse• documented• compa`ble

O(105)d.o.f.16

•  Observable:globallyaveragedTS•  Forcing:increaseofCO2concentra`on•  Linearresponse:•  Weperformensembleexperiments

– Concentra`onatt=0•  Fantas`c,wees`mate

•  …andwepredict:

Step1

T S f

(1) t( ) = dσ∫ GTS(1) σ( ) f t −σ( )

ddt

T S f

(1) t( ) = εGTS(1) t( )

f t( ) = εΘ t( )

T S g

(1) t( ) = dσ∫ GTS(1) σ( )g t −σ( )

f t( )

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ClimateChangePredic5on-TS

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[CO2]360ppm→720ppmat1%peryear2Xayerτ≈70years,constantayerthat

T S gτ

(1) t( ) = dσ∫ GTS(1) σ( )gτ t −σ( )

Predic5on

(Transient)ClimateSensi5vity

•  ΔTattheendoftheramp(τ=70ys)•  SmallerthanECS(4.1Kvs4.8K)•  Wanttodefineiner`aatallvaluesofτ

–  Instantaneousvsquasista`c19

TCR τ( ) = T S gτ

(1)τ( ) = dσ∫ GTS

(1) σ( )gτ τ −σ( )

= ECS −P fCO2

2 x χTs(1)(ω)1+ sinc(ωτ / 2)e−iωτ /2

2πiω−∞

+∞

∫ dω

ECS =ℜ χTS(1) 0( ){ }= 2

πdω∫ Re[ Ts

(1) (ω)] “EQUILIBRIUM”

“TRANSIENT”

TSProjec5on

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Error

NonlinearProcess

Ice-Albedo

BeyondGlobalIndicators

CommonSense•  Forcedfluctua`onswillprojectonthefreeones

–  FDTwillwork•  …unlessyouareinlowdimensionand/orusecooked-upobservables&forcings

•  Pastexperiments:some`mesFDTworks,some`mesitdoesnotwork.Whyso?

•  ..Butsome`mescowsarenotreallyspherical...

Gritsun&Lucarini,2017,PhysicaD21

A

PhysicaD349,62-76(2017)

Simplemodelofthemid-la`tudes

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•  Ψisstreamfunc`on,ΔΨvor`city•  Rota`on,Orographicforcing,diffusion,fric`on•  Externaldriving•  Madetolooklikewinteratmosphere•  Verychao`c(#(λj,>0)=28,231dof)

ResponsetoForcings•  Responseorographicforcingvsnaturalvariability•  ResonanceinexplicablewithFDT

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•  ResonancecomesfromagroupofUPOs•  UPOsrarelyvisitedbysystembutresonant•  Possibleparadigmforclima5csurprises

EffectofOrographicforcing

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“Tippingelements”

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•  “Highlysensi`ve”regionstoclimatechange,“irreversible”responsetoforcings

Lentonetal.2007

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HighSensi`vityvsTippingPoint

28StolenfromLenton,somewhere…

Effec5vePoten5alplusnoise

Cold

Warm

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•  Asimplemodeloftheform•  Transi`ons:noiseac`ngontheeffec`vepoten`al•  Prob≈exp[-2ΔV/ε2]•  Timeseriesanalysisvsdynamics:Procedureisnotrobust

•  Unless:EVT-Faranda,Lucarini,Manneville,Wouters2014

e.g.Lentonetal.2008

dY = −dV dY +αdW

ΔV

ΔV

ClimateResponse

A.  Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?

B.  Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters

C.  Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor

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B

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B

WhathappensNearCri`calTransi`ontwoseparateprocesses:•  Cri`calSlowingdown:thedecayofcorrela`onsbecomesslowerandslower– PropertyoftheaFractor– Thesystemhaslongermemory– Theresponsetoperturba`ondiverges– RadiusofexpansionofRuelle’stheory

•  Convergenceofensembles:theaFractoraFractslessefficientlynearbytrajectories– PropertyofneighborhoodoftheaFractor– CannotbeflaggedbydynamicsontheaFractor

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ClimateResponse

A.  Smoothresponse–ResponseTheoryConstruc`ngthesensi`vityoftheclimateandthemeasureofthepullbackaFractor.Linkbetweenclimatevariabilityandclimatechange?

B.  Highsensi`vity–Ruelle-PollicoFResonancesRoughdependenceonsystem’sparameters

C.  Cri`calTransi`onsCrisisofthehigh-dimensionalaFractor

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Snowball/SnowfreeTransi5ons

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•  …andthereservedtransi`on

HistoricalNote

•  ThebistabilityoftheEarthsystemwasdiscoveredwhenstudyingthepossibleeffectsofthenuclearwinter

•  Budyko,Sellersinthelate‘60realizedthataprolongednuclearwintermightleadtoaglobalglacia`on

•  Ghil(1976)extendedtheanalysis•  Peoplelaughedatthispossibility…butinearly’90spaleoevidencesemerged!– Bewarecri`caltransi`ons!

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Feedbacks(1)

•  Radia`vefeedback•  Warmerbodiesemitmore

•  Coolerbodiesemitless

•  Nega`vefeedback

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Feedbacks(2)•  Ice-albedofeedback:•  Awarmersurfacehaslesssnowcover

•  Albedodecreases•  Moreradia`onisabsorbed

•  Temperatureanomalyisstrengthened

•  Posi`vefeedback38

0-DEnergybalance

Ice/AlbedoFeedback

•  Energybalance:

•  With:

39 100 150 200 250 300 350 4000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

T (K )

α p T( )

α1, T < T1

α1 +T −T1

T2 −T1

α2 −α1( ), T2 < T < T1

α2 T > T2

"

#$$

%$$

C dTdt

= I −O

C dTdt

= 1−α p T( )( ) S4 − A+BT( )

I0

Bifurca5ons

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Solar Irradiance

Surface

Tem

peratu

re

Bistability

TIPPINGPOINT

TIPPINGPOINT

STABLE

STABLE

UNSTABLE

1DEnergyBalanceModel–Ghil‘76

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DiffusionparameterGreenhouseeffect

Ice-albedofeedback

TemperatureProfiles

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Grebogi,O^,Yorke(1983):

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A

?

Fromthis..

…tothis!

?

Clima5cEdgeStates

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CO2 S*

Boschietal.2013

DIFF

NonDIFF

EDGESTATES

45

C Clim.Dyn.44,3361-3381(2015)

46

C

Nonlinearity30,R32-R66(2017)

SpectralAtmospheremoistprimi`veequa`ons

onσlevels

Sea-Icethermodynamic

TerrestrialSurface:fivelayersoilplussnow

Vegeta`ons(Simba,V-code,

Koeppen)

Ghil-SellersDiffusiveOceanModelwithAlbedo

PUMA-GS

•  SimplifiedClimateModel

•  Primi`veEqua`onsAtmosphere

•  SimpleDiffusiveGhil‘76Ocean

•  Slowandfastclimatevariability

•  Posi`ve/nega`vefeedbacks

O(105)d.o.f.

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EdgeState:“rela5vea^ractor”onthebasinboundary

481

2

3

4

5

W

C

E

ixw,0,0

xc,0,0

"1

•  AyerGOY1983:Eckhardtetal.,mul`stablefluids– Pipeflow,PlaneCoueFeFlowwithfixedpointvs(transient)turbulentregimeforsuitableRe

•  Dynamicsonthebasinboundary

TrackingtheEdgeState

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•  Dynamicsofanorbitnearthethebasinboundary–  First,itrelaxesVERYrapidlytowardstheedgestate;–  Second,itdecidestowardswhichaFractoritshouldheadto;–  Third,itreachesthefinaldes`na`on.

•  Reitera`ngtheprocedureweenduponthe..“Melancholiastates”(ayerL.VonTrier’smovie)

Acloserlookattheboundary•  Pikovsky:“isthebasinboundarysmooth?”

–  Itisfolded,indeedfractal.– Resultof1024integra`onsbetweentwotrajectoriesneartheboundary,0.5KdifferenceinTsurf

– Resultofdifferent`mescalesofinstabilityontheedgestatesvsacrossit

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Mul5pleSteadyStates

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CHAOTICMELANCHOLIASTATE

CHAOTICWARMCLIMATESTA

TE

STATIONARYCOLDCLIMATESTATE

SYMMETRYBREAK

SYMM.BREAK

CHAOTICTRANSIENTSTATE

3CLIMATES

3STABLESTATES

Mul5stability

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ColdState MelancholiaState WarmState

SymmetryBreakoftheMelancholiaState

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Conclusions•  Climateasnonequilibriumsta`s`calmechanicalsystem•  Beyondinvariantmeasure:pullbackaFractor•  Responsetheoryforsmoothresponse

–  Predictclimatechange,•  Highsensi`vityandmixingrate

–  Transferoperatorapproach•  Mul`stabilityandTippingpoints

– MelancholiaState,gateforthetransi`ons•  WecanconstructtheMelancholiastate,whichseparatesthewarm

fromthesnowballclimatestate,alsoinrealGCMs•  Theedgestateisthegateallowingfornoise-inducedtransi`ons

betweenthetwobasinsofaFrac`on•  Note:ProximitytoTippingPointscanbedetectedusingEVT(Faranda,

Lucarini,Manneville,2014)

BewareCri`calTransi`ons!